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A341809 Number of ways to write n as an ordered sum of 10 nonzero tetrahedral numbers.

%I A341809 #9 Sep 08 2022 08:46:26
%S A341809 1,0,0,10,0,0,45,0,0,130,0,0,300,0,0,612,0,0,1095,10,0,1740,90,0,2565,
%T A341809 360,0,3490,930,0,4351,1980,0,5130,3790,0,5680,6330,45,5820,9540,360,
%U A341809 5715,13620,1260,5292,17950,2880,4530,22140,5670,3780,26490,10170,2940,29770,15840
%N A341809 Number of ways to write n as an ordered sum of 10 nonzero tetrahedral numbers.
%H A341809 G. C. Greubel, <a href="/A341809/b341809.txt">Table of n, a(n) for n = 10..1000</a>
%F A341809 G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^10.
%t A341809 nmax = 66; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^10, {x, 0, nmax}], x] // Drop[#, 10] &
%o A341809 (Magma)
%o A341809 R<x>:=PowerSeriesRing(Integers(), 70);
%o A341809 Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..70]])^10 )); // _G. C. Greubel_, Jul 18 2022
%o A341809 (Sage)
%o A341809 def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..8) ) )^m
%o A341809 def A341809_list(prec):
%o A341809     P.<x> = PowerSeriesRing(ZZ, prec)
%o A341809     return P( f(10, x) ).list()
%o A341809 a=A341809_list(100); a[10:71]  # _G. C. Greubel_, Jul 18 2022
%Y A341809 Cf. A000292, A023533, A023670, A282582, A340955, A341793, A341794, A341795, A341796, A341797, A341806, A341807, A341808.
%K A341809 nonn
%O A341809 10,4
%A A341809 _Ilya Gutkovskiy_, Feb 20 2021